How to Complete a Perfect Square
In mathematics, a perfect square is a number that can be expressed as the square of an integer. Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. This process not only simplifies the equation but also helps in finding the roots of the equation more easily. In this article, we will discuss the steps involved in completing the square and how to apply them to various quadratic equations.
Step 1: Write the quadratic equation in the form ax^2 + bx + c = 0
Before we can complete the square, we need to ensure that the quadratic equation is in the standard form. If the equation is not in this form, we must rearrange the terms so that the equation reads ax^2 + bx + c = 0. Here, a, b, and c are constants, and a is not equal to zero.
Step 2: Identify the value of a
Once the equation is in the standard form, identify the coefficient of the x^2 term, which is represented by ‘a’. This value will be used in the next step to complete the square.
Step 3: Divide the coefficient of the x-term by 2 and square the result
Next, divide the coefficient of the x-term (b) by 2, and then square the result. This will give us the value needed to complete the square. Let’s denote this value as (b/2)^2.
Step 4: Add and subtract the square of the result from the equation
Now, add and subtract (b/2)^2 from the left side of the equation. This will allow us to group the x-terms and create a perfect square trinomial. The equation will now look like this:
ax^2 + bx + (b/2)^2 – (b/2)^2 + c = 0
Step 5: Rewrite the equation as a perfect square trinomial
Group the x-terms and the constant terms, and rewrite the equation as a perfect square trinomial. The perfect square trinomial will be of the form (x + k)^2, where k is the value obtained in step 3.
ax^2 + bx + (b/2)^2 = (b/2)^2 – c
(ax + b/2)^2 = (b^2 – 4ac) / 4a
Step 6: Solve for x
At this point, we have transformed the quadratic equation into a perfect square trinomial. To find the roots of the equation, take the square root of both sides and solve for x.
x + b/2 = ±√((b^2 – 4ac) / 4a)
x = -b/2 ± √((b^2 – 4ac) / 4a)
By following these steps, you can complete a perfect square and solve quadratic equations more efficiently. Remember to always double-check your work and ensure that the equation is in the correct form before proceeding with the process.
